Question

If \[ \sin\left(\frac{\pi}{18}\right)\sin\left(\frac{5\pi}{18}\right)\sin\left(\frac{7\pi}{18}\right)=K, \] then the value of \[ \sin\left(\frac{10K\pi}{3}\right) \] is:

• A. \( \dfrac{\sqrt3+1}{2\sqrt2} \)
• B. \( \dfrac{\sqrt3-1}{\sqrt2} \)
• C. \( \dfrac{\sqrt3}{2} \)
• D. \( \dfrac{1}{2} \)

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The given expression is a product of sine functions with specific angles. We can convert these radian measures to degrees for easier manipulation and use trigonometric product-to-sum identities or specialized sine product formulas.
Step 2: Key Formula or Approach:
1. Radian to Degree: \(\frac{\pi}{18} = 10^\circ, \frac{5\pi}{18} = 50^\circ, \frac{7\pi}{18} = 70^\circ\).
2. Identity: \(\sin \theta \sin(60^\circ - \theta) \sin(60^\circ + \theta) = \frac{1}{4} \sin 3\theta\).
Step 3: Detailed Explanation:
Let the product be \(K\):
\[ K = \sin 10^\circ \sin 50^\circ \sin 70^\circ \]
Using the identity with \(\theta = 10^\circ\):
\[ K = \sin 10^\circ \sin(60^\circ - 10^\circ) \sin(60^\circ + 10^\circ) = \frac{1}{4} \sin(3 \times 10^\circ) \]
\[ K = \frac{1}{4} \sin 30^\circ = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \]
Now, we need to find the value of \(\sin \left(\frac{10 K \pi}{3}\right)\):
\[ \sin \left(\frac{10 \times (1/8) \times \pi}{3}\right) = \sin \left(\frac{10\pi}{24}\right) = \sin \left(\frac{5\pi}{12}\right) \]
Converting back to degrees: \(\frac{5\pi}{12} = \frac{5 \times 180^\circ}{12} = 75^\circ\).
Using the value of \(\sin 75^\circ\):
\[ \sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \]
\[ = \left(\frac{1}{\sqrt{2}}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{3}+1}{2\sqrt{2}} \]
Step 4: Final Answer:
The value is \(\frac{\sqrt{3}+1}{2 \sqrt{2}}\).